What is structure? The definitions of structure that I've read define it "a set containing mathematical objects endowed with an operation that enriches its structure."
The inclusion of the word being defined in the definition is confusing. I've read several articles on structure - and as far as I can tell, it just denotes 'a set containing mathematical objects and some operations'. Is my definition of structure adequate? 
 A: Your definition seems good for the algebraic structures, but note that we can have also more sets and more operations, as in the case of  vector spaces or of  modules.
Also, we can have ''structures'' defined by means of a relation (not an operation), as an ordered set, or adding to a set ( or to an underlying structure)  some extra element, as a family of subsets with special properties (as in the case of topological or metric spaces) or a function with suitable properties (as a metric space or a manifold) .
A: Emilio also gave other important examples of structures. In general I don't think one can say more than: a structure is a tuple of sets. Some of these sets may be "underlying sets", some functions and others relations (functions and relations are somewhat interchangable anyway).
However, to understand what really "makes a kind of structure work" you need to ask yourself what the homomorphisms of this kind of structure are supposed to look like. This will reflect what you really care about in your structure.
